Mapping determination methods and data discrimination methods using the same

ABSTRACT

A mapping determination method for obtaining mapping F from an N-dimensional metric vector space Ω N  to an M-dimensional metric vector space Ω M  has the following steps to get the optimal mapping quickly and positively. In the first step, complete, periodic, L m  basic functions g m  (X) according to the distribution of samples classified into Q categories on the N-dimensional metric vector space Ω N  are set. In the second step, a function f m  (X) indicating the m-th component of the mapping F is expressed with the linear sum of the functions g m  (X) and L m  coefficients c m . The third step provides Q teacher vectors T q  =(t q .1, t q .2, t q .3, . . . , t q .M) (where q=1, 2, . . . , Q) for the categories on the M-dimensional metric vector space Ω M , calculates the specified estimation function J, and obtains the coefficients c m  which minimize the estimation function J. In the fourth step, the coefficients c m  obtained in the third step are stored in memory.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to mapping determination methods required in various fields, such as pattern recognition, pattern generation, control systems for industrial robots, and prediction processing systems for economic problems and, more particularly, to a mapping determination system for implementing the desired mapping effectively and efficiently with the required precision and with an estimation function being minimized when the features of the mapping are determined by learning.

2. Description of the Related Art

In many fields, such as pattern recognition including voice recognition and image recognition, pattern generation including voice generation and computer graphics, prediction processing systems for economic prediction and stock-price prediction, and control systems for industrial robots, processing systems having mapping which generates or outputs output vectors in a specified dimension from input vectors in a specified dimension are used.

In voice recognition and image recognition, for example, linear mapping or nonlinear mapping is used for compressing characteristic vectors obtained from input data in digital signal processing.

Orthogonal transformations such as discrete Fourier transformation (DFT) are linear mathematical transformations. A logarithmic transformation is a nonlinear mathematical transformation. Since these mappings are fixed, however, they are hardly applied to a system in which the desired output vectors are obtained from any input vectors.

Therefore, methods for determining the desired mapping by learning has been examined. Typical examples of this kind of mapping are the KL transformation (linear) and a hierarchy neural network (nonlinear). More particularly, the hierarchy neural network has been applied to many fields because it can theoretically express any continuous mapping by increasing the number of intermediate layers.

This hierarchy neural network is a network in which connections are made from the input layer to the output layer such that the outputs of basic units (neuron elements) of each layer are connected to the inputs of basic units of the next layer. FIG. 1 shows an example of such a hierarchy neural network. In this example, the neural network includes three layers. The input layer has four elements (elements 1 to 4), the intermediate layer has three elements (elements 5 to 7), and the output layer has one element (element 8).

Processing performed in a general three-layer neural network having N elements in the input layer, L elements in the intermediate layer, and M elements in the output layer will be described below. Assume that an input vector X input to the input layer is expressed as X=(x₁, x₂, x₃, . . . , x_(N)) output vector Y output from the corresponding output layer is expressed as Y=(y₁, y₂, y₃, . . . , y_(M)).

The outputs from the N elements in the input layer are simply the inputs x_(i) (i=1, 2, 3, . . . , N) to the elements, and they are input to the L elements in the intermediate layer. The intermediate layer performs the following calculation and outputs the results. ω_(ij) indicates combination weight coefficients and s(x) is a sigmoid function. ##EQU1## where, j=1, 2, . . . , L

X_(N) ω_(Nj) =ω_(Nj)

The outputs x'_(j) (j=1, 2, 3, . . . , L) from the L elements in the intermediate layer are input to the M elements in the output layer. The output layer performs the following calculation. ##EQU2## where, k=1, 2, . . . , M

X'_(L) ω'_(kL) =ω'_(kL)

A neural network having four or more layers basically has the same network structure except for increases in the number of layers depending on an input-output relationship.

In a neural network, mapping is restricted to this kind of structure. In order to set the features of the mapping (to set combination weight coefficients ω), learning samples are given to the input layer and the outputs of the mapping (the neural network) corresponding to the learning samples are obtained. Then, teacher vectors are given corresponding to these mapping outputs, an estimation function is specified as the sum of square errors between the mapping outputs and the teacher vectors, and the combination weight coefficients ω) are determined by back propagation. The algorithm of this back propagation is a steepest descent method (probability descent method) for each data applied to a neural network.

In the steepest descent method, the result generally depends on how initial values are given to an estimation function, depending on the type of the estimation function. A solution corresponding to a minimal (local minimum) value may be obtained instead of the optimal solution corresponding to the minimum (global minimum) value.

The estimation function in the neural network exactly falls in this case. It is not assured that the solution obtained in back propagation has the minimum error. This means that depending on given initial values mapping may be obtained with outputs quite different from the teacher vectors.

Therefore, in order to obtain the best possible quasi-optimal solution, measures are taken such as a method in which learning is repeated with random numbers given as initial numbers.

It is not assured, however, that the optimal solution (the minimum value) will be theoretically obtained with these symptomatic treatments. Quasi-optimal solutions may have larger errors compared with the minimum error of the optimal solution. It takes a huge amount of learning time to get even such quasi-optimal solutions.

In addition, only when the number of neuron elements in an intermediate layer is indefinite, namely, in a ideal condition, can the neural network express any continuous mapping. In reality, the desired mapping is configured with an intermediate layer having the definite given number of neuron elements. In other words, the performance of a neural network is the extent to which an ideal mapping is approximated when the number of elements in an intermediate layer is limited to a number in actual use.

The degrees of structural freedom in a neural network comprise the number of hierarchial layers and the number of elements, both of which affect the size of the network, as well as combination weight coefficients. Therefore, a neural network may have insufficient approximation capability under the limited size in an actual use.

As described above, a neural network serving as a conventional learning-type mapping apparatus has the following three problems in learning.

(1) A minimum error is not assured for solutions. Solutions may be locally minimum values.

(2) A great amount of learning time is required to obtain a solution as close as possible to the optimal solution.

(3) A neural network may have insufficient approximation capability for the desired mapping in relation to the size in actual use.

SUMMARY OF THE INVENTION

Under these conditions, the present invention is made in order to get the optimal solution quickly and positively and to obtain mapping having higher approximation capability.

Accordingly, it is an object of the present invention to provide a mapping determination method especially suitable for periodic mapping, wherein the desired mapping is given under the condition that the estimation function is minimized and a great amount of learning time is not required to determine the mapping.

The above object of the present invention is achieved through the provision of a mapping determination method for obtaining mapping F from an N-dimensional metric vector space Ω_(N) to an M-dimensional metric vector space Ω_(M), comprising: a first step of setting complete, periodic, L_(m) basic functions g_(m) (X) according to the distribution of samples classified into Q categories on the N-th dimensional metric vector space Ω_(N) ; a second step of expressing a function f_(m) (X) indicating the m-th component of the mapping F with the linear sum of the function g_(m) (X) and L_(m) coefficients c_(m) ; a third step of providing Q teacher vectors T_(q) =(t_(q).1, t_(q).2, t_(q).3, . . . , t_(q).M) (where q=1, 2, . . . , Q) for the categories on the M-dimensional metric vector space Ω_(M), calculating the specified estimation function J, and obtaining the coefficients c_(m) which minimize the estimation function J; and a fourth step of storing the coefficients c_(m) obtained in the third step.

The estimation function J may be expressed as follows, where E{XεS_(q) }×{f(X)} indicates a calculation for obtaining the expected values of the function f(X) over all elements in a learning sample S_(q). ##EQU3##

The coefficients c_(m) which minimize the estimation function J may be obtained by partially differentiating the estimation function J by the coefficients c_(m), and setting the result to 0 in the third step.

The basic functions g_(m) (X) may be trigonometric functions.

The above object of the present invention is also achieved through the provision of a data discrimination method for discriminating input-data categories using the mapping function obtained from the above-described mapping determination method.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates a configuration example of a neural network.

FIG. 2 shows a configuration of a network using a mapping determination method of the present invention.

FIG. 3 is a flowchart indicating the processes of the mapping determination method of the present invention.

FIG. 4 is a block diagram showing a configuration example of a discrimination apparatus to which the mapping determination method of the present invention is applied.

FIG. 5 indicates two-dimensional learning data in categories C₁ and C₂.

FIG. 6 indicates the outputs of the mapping to which the learning data is input.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

In the present invention, a function f_(m) (X) for the m-th component of mapping F is defined as follows with linear combination of L_(m) functions g_(m).1 (X) when the mapping F from an N-dimensional metric vector space Ω_(M) to an M-dimensional metric vector space Ω_(M) is determined.

    f.sub.m (X)=c.sub.m.sup.T g.sub.m (X)                      3

where,

XεΩ_(N),

c_(m) = c_(m).1, c_(m).2, . . . , c_(m).Lm !^(T),

g_(m) (x)= g_(m).1 (x), g_(m).2 (x), . . . , g_(m).Lm (x)!^(T),

T: inversion

X=(x₁, x₂, x₃, . . . , x_(n)),

c_(m) : Specified coefficients.

That is, a complete function system on an N-variable function space is employed as the function g_(m).1 (x) in the present invention. From a theorem in function analysis that says any function can be expressed with linear combination of complete function systems, it is understood that any continuous mapping can be theoretically expressed with the functions g_(m).1 (x) when the number L_(m) is large enough. This corresponds to the condition that any continuous mapping can be theoretically expressed when the number of neuron elements in the intermediate layer of a hierarchy neural network is large enough.

For comparison with the neural network shown in FIG. 1, a network according to mapping of the present invention is illustrated in FIG. 2. Inputs x₁ to x₄ are input to elements 11 to 14, then are output to elements 15 to 17 in the intermediate layer as is. The element 15 performs the calculation shown by the following expression.

    X'.sub.1 =c.sub.1 g.sub.1 (X)                              4

The function g₁ (X) (=g₁ (x₁, x₂, x₃, x₄)) is calculated from the variables x₁, x₂, x₃, and x₄. Then coefficient c₁ is multiplied. In the same way, the elements 16 and 17 perform the calculations shown by the following expressions.

    X'.sub.2 =c.sub.2 g.sub.2 (X)                              5

    X'.sub.3 =c.sub.3 g.sub.3 (X)                              6

The element 18 in the output layer calculates the sum of x'₁, x'₂, and x'₃ output from the elements 15 to 17 to obtain the output y.

When the functions g_(i) (X) are selected or set to the specific functions, mapping F is obtained by setting coefficients c_(i) to the specific values in learning.

In other words, the functions g_(i) (X) are selected such that the pattern structure to be analyzed is more clearly identified. When a pattern is distributed on three classes (categories) in one dimension, the three classes cannot be identified with functions 1 and X. It is necessary to add a term of the second or subsequent order X^(i) (i>1) to the functions g_(i) (X).

To determine the coefficients C_(i), learning samples (set of the learning samples on categories C_(q) : S_(q) (=(S_(q1), S_(q2), . . . , S_(qN))) classified into Q categories C_(q) (q=1, 2, 3, . . . Q) on an N-dimensional (metric) vector space Ω_(N) are used together with Q teacher vectors T_(q) (=(t_(ql1), t_(q).2, t_(q).3, . . . , t_(q).M), where q=1, 2, . . . , Q) on an M-dimensional (metric) vector space Ω_(N) against the categories C_(q) to calculate the estimation function J expressed by the following expression. It is preferred that the teacher vectors be at general positions. This means that t_(q).1 t_(q).2, t_(q).2 t_(q).3, . . . , t_(q). M-1 t_(q).M are linearly independent. ##EQU4##

Let E{xεS_(q) } ! mean calculation of expected values (averages) over S_(q) within !. Coefficient vector c_(m) is determined in advance such that this estimation function is minimized. The desired mapping can be performed against input vectors by calculating f_(m) (x) specified in the expression 3 with the use of the obtained coefficient vector c_(m).

Based on the Hibert's famous theorem in function analysis that says any function can be expressed with linear combination of complete function systems, any continuous mapping can be theoretically expressed by employing complete function systems in an n-variable function space as the basic functions g_(m).i (x) and making the number L_(m) large enough.

How to obtain coefficient vector c_(m) which makes the estimation function, described in the expression 7, minimum will be described below. From the expression 7, the following expression can be obtained. ##EQU5##

The expression 8 can be expressed as follows; ##EQU6##

The expression 3 is substituted into the expression 10 to get the following expression. ##EQU7##

G_(M) is an L_(m) -by-L_(m) symmetrical matrix, H_(M) is an L_(m) -order vector, and K_(M) is a constant, obtained from learning samples and teacher vectors. The expression 11 is substituted into the expression 9 to get the following expression. ##EQU8##

The necessary and sufficient condition for c_(m) to minimize the expression 12 is as follows;

    ∂J/∂c.sub.m =0(m=1, 2, . . . , M) 13

From this expression, the following expression is obtained.

    G.sub.m c.sub.m -H.sub.m =)(m=1, 2, . . . , M)             14

By solving this equation for c_(m), the coefficient vector c_(m) is determined. The coefficient c_(m) for minimizing the estimation function J is obtained by solving the equation 14.

As described above, when mapping according to the present invention is determined, minimizing the estimation function is assured except for special cases such as when the equation 14 is indeterminate or inconsistent. This means, instead of solving the equation 14, applying the steepest descent method to the estimation function J, shown in the expression 12, also generates the solution c_(m) uniquely without being annoyed by the initial-value problem.

This feature, that a unique solution is obtained, eliminates repeated learning for obtaining a quasi-optimal solution in a neural network with an initial value being changed.

Assume that the mapping shown in the expression 3 uses a periodic function (f(x)=f(x+θ), θ: period), especially a trigonometric function, for the basic function. That is, the mapping in which the m-th order function is expressed by the expression 15 is used. Any periodic function is approximated by a periodic function with fewer terms than, for example, a monomial expression with N variables.

    f.sub.m (x)=a.sub.m +b.sub.m.sup.T.sub.m (x)+c.sub.m.sup.T.sub.m (x) 15

where a_(m), b_(m) = b_(m).1, b_(m).2, . . . , b_(m).Im !^(T), and c_(m) = c_(m).1, c_(m).2, . . . , c_(m).Jm !^(T) are coefficients, 1, g_(m) (x)= g_(m).1 (x), g_(m).2 (x), . . . , g_(m).Im (x)!^(T), and h_(m) (x)= h_(m).1 (x), h_(m).2 (x), . . . , h_(m).Jm (x)!^(T) are basic functions. g_(m).i (x) and h_(m).j (x) are expressed as follows;

    g.sub.m.i (x)=cos(p.sub.i.sup.T x) (i=1, . . . , I.sub.m)  16

    h.sub.m.j (x)=sin(q.sub.j.sup.T x) (j=1, . . . , J.sub.m)  17

p_(i) and q_(j) are N-dimensional vectors and selected such that the following is linearly independent.

{1, g_(m).1 (x), g_(m).2 (x), . . . , g_(m).im (x), h_(m).1 (x), h_(m).2 (x), . . . , h_(m).Jm (x)}

When xεΩ₂, that is, N=2, for example, g_(m).i (x) corresponds to cosX₁, cos2x₁, cos3x₁, . . . , cosX₂, cos2x₂, cos3x₂, . . . , cos(x₁ +x₂), . . . and so on. The sine function h_(m).j (x) corresponds in the same way.

The following expressions are defined to express the expression 15 in the same way as the expression 3.

    r.sub.m = a.sub.m b.sub.m.sup.T c.sub.m.sup.T !.sup.T      18

    s.sub.m (t)= 1 g.sub.m.sup.T (x)h.sub.m.sup.T (x)!.sup.T   19

With the expressions 13 and 14, the expression 3 can be expressed as follows;

    f.sub.m (x)=r.sub.m.sup.T s.sub.m (x)                      20

By defining the estimation function J as the expression 7, r_(m) which minimizes J can be obtained in the same way as for c_(m).

    ∂J/∂r.sub.m =0(m=1, 2, . . . , M) 21

From this expression, the following equation is obtained.

    G.sub.m r.sub.m -H.sub.m =0 (m=1, 2, . . . , M)            22

G_(m) and H_(m) are coefficients obtained from the following expressions. ##EQU9##

Mapping F which minimizes the estimation function J is obtained by solving the equation 22 to get r_(m). Being suitable for minimizing the estimation function J, this mapping is superior in terms of using a less number of terms in approximating a periodic mapping with the use of trigonometric functions as the basic functions.

FIG. 3 shows the above-described process. A basic function vector s_(m) (X) using trigonometric functions is determined according to the expression 19 in a step S1. Coefficients G_(m) and H_(m) are obtained from learning data and teacher vectors according to the expressions 23 and 24 in a step S2. Then in a step S3, a coefficient vector r_(m) is obtained by solving the expression 22. Next in a step S4, mapping f_(m) (x) is obtained from the basic function vector and the coefficient vector according to the expression 20.

As the basic function, a cosine function only, a sine function only or combinations of polynomials and trigonometric functions may be used.

An example will be described below in which the above-described mapping determination method is applied to a discrimination apparatus for classifying two-dimensional data into two categories. FIG. 4 shows the category discrimination apparatus. Sample data for determining mapping is supplied to a mapping determination section 44. It may be two-dimensional data to be input to a guaranteed global minimum mapping (GGM) calculation section 41. The mapping determination section 44 determines mapping coefficients c_(i) using the mapping determination method described above. The determined coefficients c_(i) are stored in a coefficient storage 42. The two-dimensional data is input to the GGM calculation section 41, and calculation is performed to obtain the function f_(m) (x) of mapping F. Referring to the coefficients c_(i), stored in the coefficient storage 42, if necessary, the GGM calculation section 41 performs the specified calculation. The output y of the GGM calculation section 41 is sent to a discrimination section 43 for the specified discrimination. This apparatus corresponds to a system having a two-dimensional input space (N=2), one-dimensional output space (M=1), and two categories (Q=2, that is C₁ and C₂). The configuration of the discrimination apparatus will be described below in a case in which two-dimensional learning data (x₁ and x₂) are given for two categories and T₁ =1 and T₂ =0 are given as teacher vectors (scalar) for the categories C₁ and C₂.

Assume that two-dimensional data (x₁ and x₂) shown in FIG. 5 is given. In this example, the learning data for the category C₁ exists around points (0.0, 0.0), (0.4, 0.4), and (0.8, 0.8). The learning data for the category C₂ exists around points (0.2, 0.2) and (0.6, 0.6). They are indicated by black points and white points, respectively. The number of C₁ data points is 156 and that of C₂ data points is 105, both including some points in common. This learning data seems to appear periodically. The mapping is determined by using the following 29 functions as the basic functions in f_(m) (x), shown in the expression 3.

1, cosX₁, cosX₂, cos2x₁, cos2x₂, cos(x₁ +x₂), cos3x₁, cos3x₂, cos(2x₁ +x₂), cos(x₁ +2X₂), cos4x₁, cos4X₂, cos(3x₁ +X₂), cos(2x₁ +2x₂), cos(x₁ +3x₂), sinX₁, sinX₂, sin2x₁, sin2x₂, sin(x₁ +x₂), sin3x₁, sin3x₂, sin(2x₁ +x₂), sin(x₁ +2x₂), sin4x₁, sin4X₂, sin(3x₁ +x₂), sin(2x₁ +2x₂), sin(x₁ +3x₂)

g_(m) (x) and h_(m) (x) are expressed with these sine functions and cosine functions. G_(m) and H_(m), shown in the expressions 23 and 24, are calculated for these basic functions. In other words, the expected values E{xεS₁ } ! in the expression 23 are obtained for the learning data in the category C₁, and the expected values E{xεS₂ } ! in the expression 23 are obtained for the learning data in the category C₂. The sum of these expected values is G_(m).

In the same way, the expected values E{xεS₁ } ! in the expression 24 are obtained for the learning data in the category C₁, and the expected values E{xεS₂ } ! in the expression 24 are obtained for the learning data in the category C₂. The sum of products with the teacher vectors T₁ =1 and T₂ =0 is calculated as H_(m). Then, the equation 22 is solved with the obtained coefficients G_(m) and H_(m) to get the coefficient vector r_(m) = a_(m) b_(m) ^(T) C_(m) ^(T) !^(T). With this coefficient vector and the basic functions, the mapping (expression 15) is determined. Table 1 shows the coefficients actually obtained for the learning data shown in FIG. 5, corresponding to the basic functions.

                  TABLE 1                                                          ______________________________________                                         Basic Functions and Corresponding Coefficients                                 No.         Basic function                                                                             Coefficient                                            ______________________________________                                         1           1           28.355072                                              2           cos x.sub.1 -10.587779                                             3           cos x.sub.2 -4.354881                                              4           cos 2x.sub.1                                                                               -37.078199                                             5           cos 2x.sub.2                                                                               -8.762493                                              6           cos (x.sub.1 + x.sub.2)                                                                    -20.528007                                             7           cos 3X.sub.1                                                                               -3.716895                                              8           cos 3X.sub.2                                                                               -0.696277                                              9           cos (2X.sub.1 + X.sub.2)                                                                   13.326908                                              10          cos (X.sub.1 + 2X.sub.2)                                                                   5.626294                                               11          cos 4X.sub.1                                                                               -0.782561                                              12          cos 4X.sub.2                                                                               6.456909                                               13          cos (3X.sub.1 + X.sub.2)                                                                   5.723860                                               14          cos (2X.sub.1 + 2X.sub.2)                                                                  5.728928                                               15          cos (X.sub.1 + 3X.sub.2)                                                                   -6.622603                                              16          sin x.sub.1 8.519443                                               17          sin x.sub.2 -0.534107                                              18          sin 2x.sub.1                                                                               -3.635216                                              19          sin 2x.sub.2                                                                               3.856045                                               20          sin (x.sub.1 + x.sub.2)                                                                    5.956769                                               21          sin 3x.sub.1                                                                               -0.886165                                              22          sin 3x.sub.2                                                                               -13.770971                                             23          sin (2x.sub.1 + x.sub.2)                                                                   12.679578                                              24          sin (x.sub.1 + 2x.sub.2)                                                                   -2.730276                                              25          sin 4x.sub.1                                                                               4.526975                                               26          sin 4x.sub.2                                                                               -3.533694                                              27          sin (3x.sub.1 + x.sub.2)                                                                   -17.697961                                             28          sin (2x.sub.1 + 2x.sub.2)                                                                  11.321379                                              29          sin (x.sub.1 + 3x.sub.2)                                                                   1.625174                                               ______________________________________                                    

FIG. 6 shows the outputs of a discrimination apparatus using the obtained mapping. The outputs for the data inputs in the category C₁ are black points whereas those of the data inputs in the category C₂ are white points. The horizontal axis indicates data numbers and the vertical axis indicates the outputs of the mapping for the corresponding data. The C₁ data is mapped around T₁ =1 and the C₂ data is mapped around T₂ =0, indicating that this discrimination apparatus effectively works for classifying the input data into the categories.

As described above, with the use of learning data in each category in an input vector space and teacher vectors in each category in an output space, mapping can be configured with linear combination of basic functions, especially using trigonometric functions as basic functions, such that the estimation function (expression 7) is minimized.

It is needless to say that the above-described method which uses trigonometric functions as the basic functions can be applied to create the desired mapping from Ω_(N) to Ω_(M) even when given learning data is periodic as in the above example. When an input vector space is limited to a finite space on Ω_(N) and learning data is periodic, this method is especially effective.

When data obtained in observation for a certain period is used for prediction at a future time as in a prediction processing system, if the observed data is periodic, a prediction processing system can be implemented for data other than the observed data, that is, future data, by configuring mapping having trigonometric functions described above as the basic functions with the observed data being used as learning data.

In addition, with the use of the present invention, the precision in pattern recognition can be increased, providing a highly precise pattern recognition apparatus.

According to the present invention, since trigonometric functions are used as the basic functions, minimization of an estimation function is assured and quick learning is enabled because repeated learning is not required. Mapping effective to periodic inputs can be implemented. 

What is claimed is:
 1. A mapping determination method for obtaining a mapping F from an N-dimensional metric vector space Ω_(N) to an M-dimensional metric vector space Ω_(M), comprising the steps of:a first step of setting L_(m) complete, periodic basic functions g_(m),i (X), where i is an integer in the range from 1 through L_(m), according to a distribution of samples classified into Q categories on said N-dimensional metric vector space Ω_(N) ; a second step of expressing functions f_(m) (X), where m=1, 2, . . . , N, each indicating the m-th component of said mapping F as a linear sum determined by said functions g_(m),i (X) and a set of L_(m) coefficients c_(m) ; a third step of providing Q teacher vectors T_(q) =(t_(q).1, t_(q).2, t_(q).2, . . . t_(q).M) (where q=1, 2, . . . , Q) for said categories on said M-dimensional metric vector space Ω_(M), calculating an estimation function J specified by the functions f_(m) (X). said teacher vectors, and said distribution of samples, and determining values of said coefficients c_(m) which minimize said estimation function J; and a fourth step of storing into memory said values of said coefficients c_(m) obtained in the third step, wherein said estimation function J is expressed as follows, where E{XεS_(q) }x {f(X)} indicates a calculation for obtaining expected values of said functions f_(m) (X) over all elements in learning samples S_(q) : ##EQU10## wherein said values of the coefficients c_(m) which minimize said estimation function J are obtained by partially differentiating said estimation function J by said coefficients c_(m), and setting the result to 0 in the third step, and wherein the third step further comprises the steps of:calculating ##EQU11## calculating ##EQU12## and obtaining said values of said coefficients c_(m) from G_(m) ×c_(m) -Hm=0.
 2. The mapping determination method of claim 1, wherein said basic functions g_(m),i (X) comprise trigonometric functions and polynomial functions.
 3. A mapping determination apparatus for obtaining a mapping F from an N-dimensional metric vector space ΩN to an M-dimensional metric vector space ΩM, comprising:first means for setting L_(m) complete, periodic basic functions g_(m),i (X), where i is an integer in the range from 1 through L_(m) according to a distribution of samples classified into Q categories on said N-dimensional metric vector space ΩN; second means for expressing functions f_(m) (X). where m=1, 2, . . . N, each indicating the m-th component of said mapping F as a linear sum determined by said functions g_(m),i (X) and a set of L_(m) coefficients c_(m) ; third means for providing Q teacher vectors T₁ =(t₁.1, t_(q).2, t_(q).3, . . . t_(q).M (where q=1, 2, . . . , Q) for said categories on said M-dimensional metric vector space Ω_(M), calculating an estimation function J specified by the functions f_(m) (X), said teacher vectors, and said distribution of samples, and determining values of said coefficients c_(m) which minimize said estimation function J: and fourth means for storing into memory said coefficients cm obtained by the third means, wherein said estimation function J is expressed as follows, where E{XεS_(q) } X {f(X)} indicates a calculation for obtaining expected values of said functions f_(m) (X) over all elements in learning samples S_(q) : ##EQU13## and wherein the third means further comprises: means for calculating ##EQU14## means for calculating ##EQU15## and means for obtaining said values of said coefficients c_(m) from G_(m) ×c_(m) -Hm=0.
 4. A data discrimination method for classifying input N-dimensional metric vectors into plural categories specified in advance, comprising the steps of:a first step of receiving said N-dimensional metric vectors; a second step of reading coefficients c_(m) stored in memory, said coefficients c_(m) being determined by a method comprising the steps of: a first calculation step of setting L_(m) complete, periodic, basic functions g_(m),i (X), where i=1, . . . , L_(m), according to a distribution of samples on said N-dimensional metric vector space ΩN a second calculation step of expressing functions f_(m) (X). where m=1, . . . , N. each indicating the m-th component of said mapping F as a linear sum determined by said functions g_(m),i (X) and a set of L_(m) coefficients c_(m) ; a third calculation step of providing Q teacher vectors T_(q) =(t_(q).1, t_(q).2, t_(q).e, . . . , t_(q).M) (where q=1, 2, . . . , Q) on an M-dimensional metric vector space ΩM, calculating a specified estimation function J, and obtaining values of coefficients c_(m) which minimize said estimation function J; and a fourth calculation step of storing into memory said values of said coefficients c_(m) obtained in the third calculation step; a step of obtaining said values of said functions f_(m) (X). each indicating the m-th component of said mapping F from said values of said coefficients c_(m) and said functions g_(m),i (X); and a step of classifying said N-dimensional metric vectors into said plural categories specified in advance from the calculation results obtained by substituting said N-dimensional metric vectors into said values of said functions f_(m) (X) wherein said estimation function J is expressed as follows, where E{XεS_(q) } x {f(X)} indicates a calculation for obtaining expected values of said functions f_(m) (X) over all elements in learning samples S_(q) ; ##EQU16## wherein said values of said coefficients c_(m) which minimize said estimation function J are obtained by partially differentiating said estimation function J by said coefficients c_(m), and setting the result to 0 in the third calculation step, and wherein the third calculation step further comprises the steps of: calculating ##EQU17## calculating ##EQU18## and obtaining said values of said coefficients c_(m) from G_(m) ×c_(m) -H_(m) =0. 